lecture 5 quantitative description of the xquantitative ... · an x-ray tube produces partially...

Lecture 5Quantitative description of the x ray vacuumQuantitative description of the x-ray vacuum

tube, synchrotron, X-FEL and X-HHGOutline

• Qualitative description of X-tubes. Quantitative description of x-ray vacuumtubes Characteristic radiation lines by the Bohr model of an atom Characteristictubes. Characteristic radiation lines by the Bohr model of an atom. Characteristiclines by Quantum Mechanics based on probabilistic wavefunction y(r,t).Probability densities corresponding to ψnlm(r) of an electron in a Hydrogen atom.

• Photon absorption and emission are described quantitatively by Einstein’s A andp q y yB coefficients. Energies of x-ray emission lines of Cu atoms. Electron bindingenergies [eV] for Cu atoms in their natural forms (relevant to lines K, L,M,…).

• The use of Van Cittert-Zernike theorem for an x-ray vacuum tube.• Synchrotrons and free-electron lasers (FELs). The three basic forms of x-ray

radiation from relativistic electrons. Quantitative description of x-rays producedby synchrotrons with a banding magnet (BM). Radiation of x-rays in narrowforward cone by relativistic electrons of synchrotrons with BMforward cone by relativistic electrons of synchrotrons with BM.

• X-ray radiation spectrum emitted by the synchrotron with BM. The use of VanCittert-Zernike theorem for a synchrotron-BM source of x-rays.

••

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 1

Outline ctd.• Narrow cone x-ray radiation generated by relativistic electrons traversing periodic

ti t t ( d l t ) Q tit ti d i ti f d d b timagnetic structure (undulator). Quantitative description of x-rays produced by motionof electrons in an undulator (A), (B) and (C). The use of Van Cittert-Zernike theoremfor x-rays produced by a synchrotron undulator. Transversely coherent x-rays withpinhole spatial filtering. Airy patterns at 500 eV and 600 eV. High transverse (spatial)pinhole spatial filtering. Airy patterns at 500 eV and 600 eV. High transverse (spatial)coherence of x-rays. High transverse (spatial) coherence of x-rays produced by spatialfiltering ALS-undulator. Comparison of synchrotrons with a bending magnet, wigglerand undulator radiations.

• X-rays from Fee Electron Lasers (FELs). Operating regimes of X-ray Fee ElectronLasers (XFELs). Typical Parameters of the accelerators of X-FELs and x-ray radiation.

• Toward the tabletop x-ray free electron lasers via a plasma based accelerator. The useof Van Cittert Zernike theorem for x rays produced by an X FELof Van Cittert-Zernike theorem for x-rays produced by an X-FEL.

• X-ray high-order harmonic generation (X-HHG). From visible-light to x-ray HHG.Unharmonic motion of an ionized electron under X-ray HHG (X-HHG). The cut-offphoton energy in X-HHG Trajectory of electron under X-HHG. X-HHG in hollowphoton energy in X HHG Trajectory of electron under X HHG. X HHG in hollowcapillary waveguides. Short modulation periods of the capillary wall extends phasematching from 85 eV to 160 eV. The Xe-plasma filled capillary waveguide extendsphase matching from 95 eV to 150 eV. The use of Van Cittert-Zernike theorem for x-

t d b X HHG S ti l h f d d b X HHGrays generated by X-HHG. Spatial coherence of x-rays produced by X-HHG.Understanding x-ray vacuum tubes, synchrotrons, X-FELs and X-HHG requires theory,computations and experiments.

• Problems as home assignments• Problems as home assignments• References

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 2

Qualitative description of X-tubesLet me consider vacuum tubes, synchrotrons, X-FELs and X-HHG in the context of transition from th i h t t h t

The spikes are characteristic lines (K, L, M,…)Kα

I (arb. u.)2

the incoherent x-ray sources to coherent ones

( )

Kβ

Δλ< 0.01 nm

λ1 Einf (n=inf.)

Shells: K, L, M, N, …Continuum)

2 4 6 8 10

βλmin

0E3

E4

M (n=3)

N (n=4)( )

L L

Fig. 1 Schematic diagram of x-radiationspectra from an x-ray vacuum tube (X-tube). H M th d d U 35 kV

λ (nm)2 4 6 8 10

E2 L (n=2)

Lα Lβ

Continuum radiation (bremsstrahlung)

( ) ( / ) /

Here, Mo-cathode and U = 35 kV.

K ( 1)

Kα Kβ Kγ

E (r, t) ~ e aT(t - r / c) / r

by de-accelerated electrons Fig. 2 Energy levels En and characteristic lines Kα, Kβ, Kγ, Lα and Lβ In an atom.

K (n=1)E1

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 3

hωmax = eU λmin = 2πh/eUβ γ α β

Quantitative description of x-ray vacuum tubes

Qualitative description of continuum radiation (bremsstrahlung): Einf (n=inf )

Shells: K, L, M, N, …Continuum)( g)

From Lecture 2 (47), the power per unit solid angle radiated by a de-accelerated electron is given by

2

E3

E4

inf

M (n=3)

N (n=4)(n inf.)

30

2

2221

16sin

ce

ddP

επΘ

=Ω

a

For the N electrons we have aE2 L (n=2)

Lα Lβ(1)

dPN (Θ) /dΩ ~ N sin2 Θ

For the N electrons, we have a Lambertian source with

Kα Kβ Kγ(2)

Fig. 4 Energy levels and characteristic lines K L M and N

Lambertianx-ray source

Θ

K (n=1)E1

Fig 3 Th L b ti

lines K, L, M and N.x-ray source

+ AnodeCathode e-beam

How can we describe quantitativelycharacteristic lines (K, L, M,…) ?


Fig. 3 The Lambertian x-ray source via a vacuum tube

Characteristic radiation lines by the Bohr model of an atommodel of an atom

Characteristic lines can be described quantitatively by using the semi-quantum Bohr model (simplest quantum model). For an example, let us consider the characteristic lines by using the ( p q ) p , y gBohr model of a Hydrogen-like ion having the nuclei charge +eZ. Assuming the stationary orbits and equating the Coulomb force Ze2/4πε0 r2 to the centripetal force mv2/r, we get

42 1emZE = (3)

Quant (hω) -e u

2220

232 nEn

hεπ=

Znan

Zr Bohrn

22

2

204

==hεπ

(3)

(4)

+

e u

l

ZmZe Bohrn 2

Using the Plank Einstein photon concept (E =h ) and the

nmaBohr 0529.0=where

(4)

(5)

++eZ

⎞⎛⎞⎛

Using the Plank-Einstein photon concept (Eph=hω) and the energy conservation law, we get the characteristic lines (photon energies)

Fig. 5 Bohr model of a Hydrogen-like atom

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=−= 2222

02

42 11

32 luluul nn

meZEEh

hεπ

ω (6)


Characteristic lines by quantum mechanics based on the probabilistic wavefunction ψ(r t)based on the probabilistic wavefunction ψ(r,t)The simplest (spin-less, non-relativistic) wave mechanics describes quantitatively the lines using the Schrödinger wave-equation

⎤⎡t

tittVm ∂

∂−=⎥

⎦

⎤⎢⎣

⎡+∇−

),(),(),(2

22 rrr ψψ hh

(7)

where rrrrr dttdtP ),(),(),( ψψ ∗=

and rrrrrrrr dttdtP ),(),(),( ψψ ∗∫∫ ==

(8)

(9)rrrrrrrr dttdtP ),(),(),( ψψ∫∫have the conventional (Copenhagen) meanings of the probability density and the expected coordinate <r> of the electron. The stationary solutions of Eq. (7) for a Hydrogen-like ion yield the stationary wavefunctions

( )

Hydrogen like ion yield the stationary wavefunctions)(rnlmψ

where n, l and m are quantum numbers. The symmetry laws allow the transitions with1±=Δl

(10)

That yields the characteristic lines (energy levels, photon energies and allowed transitions). Notice, Dirac relativistic equation takes into account also the electron spin s:

1,01

±=Δ±Δ

ml

(11)(12)


, q p2/1±=Δs

Probability densities corresponding to ψnlm(r) of an electron in a Hydrogen atomof an electron in a Hydrogen atom

Fig. 6 Calculated probability densities corresponding to the wavefunctions of an electron in a hydrogen atom possessinghydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: n = 1, 2, 3, ...) and angular momenta (increasingangular momenta (increasing across from left to right: s, p, d, ...). The probability densities are through the xz-plane for the electron at different quantumelectron at different quantum numbers (ℓ, across top; n, down side; m = 0). Brighter areas correspond to higher probability density in a positiondensity in a position measurement.(from Wikipedia)


Photon absorption and emission are described quantitatively by Einstein’s A and B coefficientsquantitatively by Einstein s A and B coefficients

The Einstein coefficient Aul, which is the inverse lifetime

Spontaneousemission

u

l

by AulThe Einstein coefficient Bul

τ of the transition u l , is calculated by QM.Stimulated

emissionul

lby Bul

ulIs calculated by QM.

ulA 1=

Absorption ul

lby Aul

ulτ

luu

lul f

gg

mceA ⎟⎟

⎠

⎞⎜⎜⎝

⎛= 3

0

22

2πεω

3

32

ωπh

cAB ulul =

Absorption and emission of a photon involves “oscillation” of an electron between. u

.)

1 l 1

ug ⎠⎝0 ωh

oscillation of an electron between the stationary states (up) and (low) at the frequencyωul = (Eu-El)/h. Stimulated emission ~ spontaneous emission induced in the predictable manner by an external EM ( h t )

tplitu

de(a

rb

0

1

roba

bilit

y

l

u

1

EM wave (photon).

Fig. 7 Oscillation amplitude and probability under the spontaneous emission u l

Osc

ill.a

mp Pr


spontaneous emission u l

Energies of x-ray emission lines of Cu atoms 1Lambertianlines K, L,M,… Lambertian

x-ray source

Cu anode

ΘCathode e-beam

22

Fig. 8 Transmissions that give rise to the various emission lines.

(Fig. 8 and Tables 1 and 2 fromhttp://xdb.lbl.gov)


Electron binding energies [eV] for Cu atoms in their natural forms (relevant to lines K L M )their natural forms (relevant to lines K, L,M,…)

3Lambertianx-ray sourceΘ

Cathode e beam

lines K, L,M,…

Note that the binding energies for Cu are used for quantitative modeling

Cu anodeCathode e-beam

of X-tubes having the anode Cu atoms

Fig. 9 Transmissions that give rise to the various emission lines.

(Fig. 9 and Table 3 from http://xdb.lbl.gov)


The use of Van Cittert-Zernike theorem for an x-ray vacuum tuberay vacuum tube

2R dOutput aperture of an x-ray tube

Fig. 10 Schema for the use of theorem in case of an x-ray vacuum tube (2-Dimensional, circular, radius R) source composed from incoherent uncorrelated emittersZ

Θ

of an x-ray tube

incoherent, uncorrelated emitters

An x-ray tube produces partially coherent or fully transversally coherent x-rays (waves) in the region.in the region.

For a circular (radius R) surface of incoherent (uncorrelated) emitters of an X-tube

ΘRkJ )(2 1 (13)Θ

=Rk

)(112μ

Thus the use of the Van Cittert-Zernike theorem for an x-ray tube yields

(13)

(14)(15)

Transverse coherence

- Incoherent radiation: 2R >> <λ>Z / d- Partially coherent radiation: 2R ~ <λ>Z / d


( )(16)

- Partially coherent radiation: 2R <λ>Z / d- Coherent radiation: 2R << <λ>Z / d

Synchrotrons and free-electron lasers (FELs)Synchrotrons and FELs are not LASERS based on amplified spontaneous emission. Nevertheless, let us briefly consider them in the context of transition from traditional incoherent x-ray sources to coherent ones

Crookes tubes X-ray vacuum tubes Synchrotrons and free-electron lasers (FEL)

Synchrotron Free electron “laser” (FEL)X-rays by acceleration of free electrons

Qualitative description of synchrotrons and FELs:

E (r, t) ~ Σi ei aT(t - r/c + ϕ) / r (17)

Fig. 11 Synchrotron (a) and FEL (b): (a)-photos from Wikipediaand (b) photos from http://flash desy de)

(a) (b)How can we describe quantitatively synchrotrons and FELs?


and (b) photos from http://flash.desy.de).and FELs?

The three basic forms of x-ray radiationfrom relativistic electronsfrom relativistic electrons

γ =(1-v2/c2)-1/2

a.u.

)dΘ ∼ 1 / γe

Banding magnet (BM)radiation

Inte

nsity

(a

h [ ]

e

hω [a.u.]

y (a

.u.)

dΘ >> 1 / γe Wiggler radiation

Inte

nsity

hω [a.u.]

sity

(a.u

.)dΘ ∼ 1 / γΝ1/2

Undulator radiation

Inte

ns

hω [a.u.]

e Undulator radiation

FEL


Fig. 13 Three forms of x-ray radiation from relativistic electrons

Quantitative description of x-rays produced by synchrotrons with a banding magnet (BM)synchrotrons with a banding magnet (BM)

Quantitative description of x-rays produced by free-electrons of synchrotrons is based on Maxwell’s equations and Einstein relativity for a free-electron.

222 idd ΘFrom Lecture 2 (28), we have(18)

From Einstein special relativity, k = γm0v and

dm

d vkF 0γ== Θ

v

F(19)

30

2

2221

16sin

ce

dtd

ddP

επΘ

=Ω

Tv

B

γ 0dt

mdt

F 0γThe Lorentz force for a relativistic electron in a constant magnetic field B is given by BvFL ×−= eUsing the equality F = F we get

R(19)

(20)Using the equality F = FL, we get

evBRvm

dtdvm −=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

2

00 γγ (21) k/k

a

Θ

dΘ

eBcm

eBvmR 00 γγ ≈=

Using the relation vT = v tan(dΘ), for the radiated x-rays, we get

(22)e

~sin2Ω

g T ( ), y , g

(23)Fig. 12 Movement of a relativistic electron in a constant magnetic field B.3

02

222

0

1

16)2/(sin)(

cde

mdevB

ddP

εππ

γ+Θ−

⎟⎟⎠

⎞⎜⎜⎝

⎛ Θ=

Ω

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 14Fig. 14 Quantitative description of x-rays produced by free-electrons of synchrotrons

⎠⎝

Radiation of x-rays in narrow forward cone by relativistic electrons of synchrotrons with BMrelativistic electrons of synchrotrons with BM

λ’λ

λ’v

λx

Angle dependent Doppler shifta

k’/k’

e

Θ’

~sin2Ω’

dΘ ∼ 1 / γe dΘ’

Fig 13 Radiation of x-raysk’x

k’dΘ’

Lorentz transformation leads tokx= k’x

kdΘ Fig. 13 Radiation of x rays

in a narrow forward cone by relativistic electrons of synchrotrons

k’z kz=2γk’z( )xx dk

kkkd

γγγ 21

2tan

'2'

=Θ

===Θ


xzz kk γγγ 22'2

X-ray radiation spectrum emitted by the synchrotron with BMsynchrotron with BM

Fig. 14 Synchrotron radiation spectrum emittedradiation spectrum emitted by SURF-BM at 380 MeV, 331 MeV, 284 MeV, 234 MeV, 183 MeV, 134 MeV and 78 MeV in134 MeV, and 78 MeV in comparison to a 3000 K blackbody (from http://physics.nist.gov/MajResFac/SURF/SURF/sr html)esFac/SURF/SURF/sr.html)

The use of Van Cittert-Zernike theorem for h t BM fa synchrotron-BM source of x-rays2R dOutput aperture

of a synchrotron-BM Fig. 15 Schema for the use of theorem in case of a synchrotron-BM x-ray source (2-Dimensional. circular, radius R) source composed from incoherent, uncorrelated Z

Θ

yx-ray source

emitters

A synchrotron-BM produces partially coherent or fully transversally coherent x-rays(waves) in the region(waves) in the region.

For a circular (radius R) surface of incoherent (uncorrelated) emitters of a synchrotron-BM x-ray source, we have

ΘΘ

=Rk

RkJ )(2 112μ (24)

ΘRk

Thus the use of the Van Cittert-Zernike theorem for a synchrotron-BM x-ray source yields


- Incoherent radiation: 2R >> <λ>Z / dPartially coherent radiation: 2R ~ <λ>Z / d

(25)(26)


- Partially coherent radiation: 2R ~ <λ>Z / d- Coherent radiation: 2R << <λ>Z / d

(26)(27)

Narrow cone x-ray radiation generated by relativistic electrons traversing periodic magneticrelativistic electrons traversing periodic magnetic

structure (undulator)

Relativistic (E=γm0c2)e-beam

Fig. 16 Narrow cone undulator x-ray radiationgenerated by relativistic

l t t i

γ =(1-v2/c2)-1/2

λ (λ /2 2)(1+K2) electrons traversing a periodic magnetic structure(from http://www.psi.ch/

i f l/h it k )

(N-periods)

Δλ/λ~1/Ν

Θ~1/γ*N1/2Θ

λ~(λu/2γ2)(1+K2)

swissfel/how-it-works).Θ 1/γ N

γ*=γ/(1+K2)1/2

K=eBλu)2πm0c


u) 0

Quantitative description of x-rays produced by motion of electrons in an undulator (A)motion of electrons in an undulator (A)

Quantitative description of x-rays produced by motion of free-electrons in an undulator is based on Maxwell’s equations and Einstein’s relativity for a free-electron. 222 iddP Θq y

From Lecture 2 (28), we have 30

2

221

16sin

ce

dtd

ddP

επΘ

=Ω

Tv

From Einstein special relativity, k = γm0v and

dm

d vkF γ== (29)

(28)

k γm0v and dt

mdt

F 0γ==The Lorentz force for a relativistic electron in a constant magnetic field B is given by BvF L ×−= eUsing the eq alit F F and the appro imation e get

(29)

(30)Using the equality F = FL, and the approximation v~vx we get

⎟⎟⎠

⎞⎜⎜⎝

⎛== yz

x zBdtdzeBev

dtdvm

λπγ 2cos0

(31) x

y

e

By

and

⎠⎝ udtdt λ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ux

zedzBdvmλπγ 2cos0

which yield

v ze

(32)

Fig. 17 Electron motion in an udulator.

which yield

(33)⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫

uu

ux

zdzeBdvmλπ

λπ

πλγ 22cos20


⎠⎝⎠⎝ uu

Quantitative description of x-rays produced by ti f l t i d l t (B)motion of electrons in an undulator (B)

⎟⎟⎠

⎞⎜⎜⎝

⎛= u

xzeBvm

λπλγ 2sin

20Thus (34)⎟⎠

⎜⎝ u

x λπγ

20

and ⎟⎟⎠

⎞⎜⎜⎝

⎛= u

xzeBv

λπλ 2sin

2(35)⎟

⎠⎜⎝ u

x m λπγ2 0

⎟⎟⎞

⎜⎜⎛

=zKcv π2sin

( )

(36)⎟⎟⎠

⎜⎜⎝ u

xvλγ

sin

wherecm

eBK u

2πλ

=

( )

(37)

Fi 18 A l 2 d l t 4 9 l 6 56

cm02π

is the magnetic deflection parameter. The deflection angle is given by Fig. 18 Apple-2 undulator: 4-9 m long, 6.56

cm period, 72 periods, 11mm minimum gap (from http://photon-science.desy.de).

The deflection angle is given by

zkKcv

vv

ux

z

x sinγ

===Θ (38)


z γ

Quantitative description of x-rays produced by motion of electrons in an undulator (C)by motion of electrons in an undulator (C)

The use of Eqs. (36)-(38) yields

λ =(λ /2γ2) (1+(K2/2)+γ2Θ2) (39)

Θ~1/γ*N1/2

λx (λu/2γ ) (1+(K /2)+γ2Θ ) (39)

(40)

Δλ/λ~1/Ν

(41)

(42)

λ~(λu/2γ2)(1+K2)

γ*=γ/(1+K2)1/2

=(1 v2/c2)-1/2

where(43)

(44)

Fig. 22 Power of x-rays in the central cone from h d l f h Ad d Li h S (ALS)

γ =(1-v2/c2)-1/2

K=eBλu)2πm0c

The use of Eqs (36)-(38) in Eq (28) yields

(44)

(45)

the undulator of the Advanced Light Source (ALS) (from http://ilsf.ipm.ac.ir/News/2014-03-03BeamlineOpWrkshp/ILSF_Attwood_Lec2_March2014.pdf).

( )2/1 2

2

0

2

KKIeP

ucen +

=λεγπ

The use of Eqs. (36) (38) in Eq. (28) yields

(46)

( )22πλ


( )cen

yyxxNcoh P

ddP

ΘΘ=.,

2πλ (47)

The use of Van Cittert-Zernike theorem for x-rays produced by a synchrotron undulatorproduced by a synchrotron undulator

2R dOutput aperture of a synchrotron undulator

Fig. 19 Schema for the use of theorem in case of a synchrotron-undulator x-ray source(2 Di i l i l di R)

ZΘ

(2-Dimensional. circular, radius R) source composed from incoherent, uncorrelated emitters

For a circular (radius R) surface of incoherent

A synchrotron undulator produces partially coherent or fully transversally coherent x-rays (waves) in the region.

( )(uncorrelated) emitters of a synchrotron-undulatorx-ray source, we have

ΘΘ

=Rk

RkJ )(2 112μ (47)

Thus the use of the Van Cittert-Zernike theorem for a synchrotron-undulator x-ray source yields



(48)(49)

Fig. 24 <Pcoh> from the undulator of ALS (from http://ilsf.ipm.ac.ir/News/2014-03-03BeamlineOpWrkshp/ILSF Attwood Lec2 March2014.pdf).



( )(50)

_ _ _ p )

Transversely coherent x-rays with pinhole spatial filt i Ai tt t 500 V d 600 Vfiltering: Airy patterns at 500 eV and 600 eV

Fig. 20 The ransversally (spatially) coherent Airy patterns at 500 eV (a) g y ( p y) y p ( )and 800 eV (b) with pinhole (d= 2.5 μm) spatial filtering using ALS(synchrotron-undulator source) with the magnetic undulator (λu= 80 mm, N = 55 periods with ma beamline (from http://www-inst.eecs.berkeley.edu/~rosfjord/).


y j )

High transverse (spatial) coherence of x-raysd d b i l fil i ALS d lproduced by spatial filtering ALS-undulator

Relativistic (E=γm0c2)e-beam

Interference (N periods)

Two i h l fringes(N-periods)

Undulator

pinholes

Undulatoraperture

Fig 21 High transverse (spatial) coherence of x-rays produced by spatial


Fig. 21 High transverse (spatial) coherence of x rays produced by spatial filtering ALS-undulator (The fringes are from C. Chang et al. Applied Opt., 42, 2506 (2003)).

Comparison of the synchrotrons having a bending magnet wiggler and undulatorbending magnet, wiggler and undulator

K=eBλu /2πmc

Synchrotrons with a banding magnet

Synchrotrons with i l (K 1) d l t

Synchrotrons with an undulator (K<1)a banding magnet a wiggler (K>>1) undulator

magnet structurean undulator (K<1)magnet structure

Hi h h tBroad spectrumHigh x-ray photon intensity

Higher x-ray photon energiesHigher x-ray photon intensity

Higher x-ray photon intensityPartial coherencydue to a small spot size


X-rays from Fee Electron Lasers (FELs)Mirror (R ~ 100%)Mirror1 (R1 100%)

Relativistice-beam

Mirror2 (R1 < 100%)

(N-periods)γ =(1-v2/c2)-1/2

TEM-likemodes

λ (λ /2 2)(1+K2)Δλ/λ~1/Ν

Θ~1/γ*N1/2

Θλ~(λu/2γ2)(1+K2)

γ*=γ/(1+K2)1/2

K=eBλu)2πm0cOperation regimes of a FEL

-Master oscillator (MO)-Amplifier-Self-amplified spontaneous radiation (SASE)


Operating regimes of the X-ray Fee Electron Lasers (X FELs)Electron Lasers (X-FELs)

O ti i f X FELOperation regimes of a X-FELs

-Master oscillator (MO): The optical pulses are traveling in MO between the resonator mirrors MO operates with smallMO between the resonator mirrors. MO operates with small gain providing a narrow bandwidth x-rays.

-Amplifier: The FEL amplifier does not have a resonator-Amplifier: The FEL amplifier does not have a resonator. Coherent seed x-ray pulses are synchronized to overlap the electron pulses.

-Self-amplified spontaneous radiation (SASE): Lasing starts via noise radiation. The x-ray wavelength may be changed by varying the el-beam energy. Mirror-less FELs , g y y g gy ,which require higher gain, are considered as the next generation of FELs.


Typical parameters of the accelerators and x-ray radiation of X-FELsray radiation of X FELs

Type of Accelerator EnergyPeak CurrentTypical parameters of accelerators Typical values of other parameters:

Peak Magnetic field: few kilogauss Wavelength: few Angstroms to 100 mm

fElectrostaticInduction LinacStorage Ring

1 - 10 MeV1 - 50 MeV100 MeV - 10 GeV

1 - 5 A1 - 10 kA1-1000 A

Number of undulator periods: 100 Undulator period λw: 2 - 10 cm Length of Undulator: 10 meters Electron beam energy: Few MeV to S l G VRF Linac 10 MeV - 25 GeV100 - 5000 A Several GeVElectron beam radius: About 1mm Electron beam pulse: nanoseconds to femtosecondsEffi i t 40 % t l

WavelengthInfrared (100 μm to millimeter)

Pulse LengthMicroseconds

Typical parameters of x-rays Efficiency: up to 40 % at longer wavelengths but less at shorter wavelengths Photon beam size (FWHM) ~ 100 μmPh t b di (FWHM) dInfrared (100 μm to millimeter)

Microns to centimetersX-ray, UV, Visible (few nanometers to micron)

MicrosecondsNanosecondsPicoseconds to

i d

Photon beam divergence (FWHM) < μradPulse duration (FWHM) ~ 100 fsMin. pulse separation ~ 90 - 100 ns Max. Number of pulses per train ~ 11500 R titi t 5 Hmicron)

X-ray to far infrared (nanometer to fraction of millimeter)

microsecondsFemtosecond to picoseconds

Repetition rate: 5 Hz Number of photons per pulse: 1.8 x 1012

Excellent beam quality M2 < 1.1 Tunability 10 GHz - 1Å


(Data from http://www.worldoflasers.com/lasertypes-electron.htm)

Toward Tabletop X-ray Free Electron Lasers b i ill l b d lby using a capillary plasma-based accelerator

Fig. 22 TowardgTabletop X-rayFree Electron Lasers by using a plasma based paccelerator. (Picture from http://www.nature.com/nphys/journal/v4/np y j2/fig_tab/nphys846_F1.html).


The use of Van Cittert-Zernike theorem for x-rays produced by an X-FELproduced by an X-FEL2R dOutput aperture

of a X-FELFig. 23 Schema for the use of theorem in case of a FEL x-ray source (2-Dimensional. i l di R) d f

ZΘ

circular, radius R) source composed from incoherent, uncorrelated emitters

For a circular (radius R) surface of incoherent (uncorrelated) emitters of a FEL x-ray source

A X-FEL produces partially coherent or fully transversally coherent x-rays (waves) in the region.

Z(uncorrelated) emitters of a FEL x-ray source, we have

ΘΘ

=Rk

RkJ )(2 112μ (51)

Relativistic

Zeff.~ τpuls c

Thus the use of the Van Cittert-Zernike theorem for a FEL x-ray source yields

e-beam

(N-periods)Θ

M1M2


- Incoherent radiation: 2R >> <λ>Zeff / dPartially coherent radiation: 2R ~<λ>Z /d

(52)(53)

2

Fig. 24 Effective distance (Z = Zeff = τpuls c)


- Partially coherent radiation: 2R ~<λ>Zeff /d- Coherent radiation: 2R << <λ>Z eff / d

( )(54)

X-ray high-order harmonic generation (X-HHG)X-ray HH Generators are not LASERS based on amplified spontaneous emission.. Nevertheless, l t b i fl id th i th t t f d l t f th t diti l h tlet us briefly consider them in the context of development of the traditional coherent x-ray sources

Femtosecond pulses are used to prevent a plasma formation. Advantages of X-HHG: The spatially coherent ultra-short (fs) pulse; Low divergent x-ray beam; The photon energy up to ~ 0.5 keV.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 31Fig. 25 Production of x-rays via HHG (Picture from KAIST, CXRC)

( ) p ; g y ; p gy p

From the visible-light HHG to the x-ray HHGQualitative descriptions using the relation (L3 (29)):

Quantitative description of coherent Quantitative description of coherent x-rays by

E (r, t) ~ Σi ei aT(t - r/c + ϕ) / rQualitative descriptions using the relation (L3 (29)):

(55)

Quantitative description of coherent visible light by coherent nonlinear motion (ϕi =ϕj ,rT << r) of bound electrons of atoms caused by the not-too-large optical field by Classic Electrodynamics:

Quantitative description of coherent x-rays by coherent nonlinear motion (ϕi =ϕj , rT << r)of bound-free-bound electrons of atoms caused by the large optical fielda (t) ~ (e/m ) E (t)optical field by Classic Electrodynamics:

aTi (t) ~ P(t) = ε0Σi χ (n) [E0(t)]n orby the perturbation approximation in Quantum Mechanics

aTi (t) (e/me) E0(t)by combination of Classic Electrodynamics and Quantum Mechanics

Unharmonicti

Nonlinear medium: hω0

En3.Recombination

X-ray

Laser

Eion

QM model motion

medium:

Gas atoms

hω

hω0

0

hω0(n) Atomic

potential

2. Accele-ration

Laser optical field

Fi 26 ( ) HHG i i ibl t l i (b) HHG i t l i(b)

f = 0E0

hω0 electron1.Tunnel ionization E0

(a)


Fig. 26 (a) HHG in visible spectral region (b) HHG in x-ray spectral region

Unharmonic motion of an ionized electronunder X ray HHG (X HHG)under X-ray HHG (X-HHG)

3 Recombination

Unharmonicmotion

3.Recombination

At i2. Accele-

X-ray

Laser optical

Eion

Atomicpotential

electron1 T l i i i

2. Acceleration

optical field

E1.Tunnel ionization E0

Three-process models are usually used for the semi-quantitative description of X-HHG.

Fig. 27 Schematic diagram of an unharmonic motion of an ionizing electron under X-HHG

1st process: Modeling of an atom ionization, which produces the quasi-free electron2nd process: Modeling of the electron oscillation caused by the laser EM field3rd process: Modeling of the electron recombination with the ion


Cut-off photon energy in X-HHGThree-process models distinguish the three basic physical processes in X-HHG, namely an atom ionization byp g p y p y ythe high-intensity laser EW-field, the electron oscillation by the high-intensity laser EM field, the electronrecombination with the ion with/without the high-intensity EM field. The ionization and recombination of anelectron in the presence the low-intensity (weak) EM field is described quantitatively by Quantum Mechanics byusing the perturbation method. Unfortunately, the QM theory based on the perturbation approximations doesg p y y p ppnot yield quantitative solutions in the case of the strong (high intensity) EM field, whose value is comparablewith the Coulomb field of the atom nuclei. Therefore, one should be satisfied by some semi-QM models, whichare very complicated. For such models, see the literature. Nevertheless, the oscillation of a free electron is welldescribed by the following simple electrodynamics model. The results are used for estimation of the cut-offy g p yphoton energy of X-HHG, which is given by

pionoffcut UE )2/3(+=−ωhwhere Eion is the ionization potential, and (3/2)Up is the quiver energy of the electron. The simple relations for the EM force acting up on a free electron

tioeeE

ddvmF ω−==

(55)

(56)of the electron. The simple relations for the EM force acting up on a free electron odt∫ −−

−== tiotio e

mieEdte

meEv ωω

ω222

222 EeEemv oio

(56)

(57)

212

21 422 ωω

ω

mEee

mEemvU o

cycletio

cyclep === −

−

−

The use L 2 (42) given by ⎟⎞

⎜⎛==I I ik

ES201 ε

yield

Example:

(58)

(59)The use L 2 (42) given by ⎟⎠

⎜⎝ k

I Intensity ES

02 μ

[ ]( ) [ ]eVmWIEeU op

22.int

142

22

1033.94

μλ⎟⎠⎞

⎜⎝⎛×== −

pUp ~ 60 eV at Iint. = 1015 W/cm, λ=1μmIn Helium: hωcut-off =Eion+3.2Up== 24.6eV+192 Ev = 220 eV

(59)

(60)


cmmp 224 ω ⎠⎝

Trajectory of electron under X-HHGTrajectory of electron

3.RecombinationX-ray

Laser

Eion

Fig. 28 Schematic diagram f j f i i d

Atomicpotential

2. Accele-ration

Laser optical field

of a trajectory of an ionized electron under X-HHGelectron

1.Tunnel ionization E0

tieeEdvmF ω−According to the model (P.B Corcum, Phs. Rev Lett (1993)), the use of th f l (61)

oeeEdt

mF ==

dtdxe

mieEe

mieEdte

meEv tioio

t

tio =−

−−

== −−−∫ 0

0

ωωττ ω

ωω

the formulas (61)

(62)

f

in

f

i

t

t

tioiot

t

tioio emi

eEemi

eEdtemi

eEemi

eEx ⎥⎦⎤

⎢⎣⎡

−−

−=⎥⎦

⎤⎢⎣⎡

−−

−= −−−−∫ 00 ωωτωωτ

ωωωωyields (63)

In the model the electron is suddenly free.The electron is released at rest from the atom (x(t0)=0).The electron trajectory ends at the atom (x(tf)=0).One solves Eq (63) for tf and finds v(tf) and return

X-HHG yields: EX-HHG (r, t) ~ Σi ei aT(t–ri/c+ϕι)/ri ~ ~dv(tf)/dt (64)


One solves Eq. (63) for tf and finds v(tf) and return electron energy E=mv2/2

X-HHG in hollow capillary waveguidesFor X-HHG, we must match the phase differences mediated by propagation of the laser EM wave in

[ ])(12 λδλπ Pk +=

For X HHG, we must match the phase differences mediated by propagation of the laser EM wave in the gas jet, where the wave number is given by (see Lecture 4 (45))

(65)

Here, k=2π/λ is the wave number in vacuum. In the case of ionized gas (gas + plasma), we have

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

πλλδ

λπ 2

21)(12 eerNPk (66)

⎦⎣ ⎠⎝ πλ 2

⎥⎤

⎢⎡

⎟⎟⎞

⎜⎜⎛

⎟⎞

⎜⎛+

λλλδπ 22 11)(12 eerNuPk

Effective phase matching in X-HHG is provided in a hollow capillary waveguide, where one get (see, Science 280, 1412 (1998))

(67)⎥⎥⎦⎢

⎢⎣

⎟⎟⎠

⎜⎜⎝

−⎟⎠⎞

⎜⎝⎛−+=

ππλδ

λ 222)(1 ee

aPk

Capillary hollow waveguide Laser wave

(67)

zWave by X-HHG Fig. 29 X-HHG in

h ll ill

Hollow capillary waveguidesallow the phase matching of the low-order harmonics.

Phase shift

a hollow capillarywaveguide (gas-filled capillary)


Short modulation periods of the capillary wall extend phase matching from 85 eV to 160 eVextend phase matching from 85 eV to 160 eVWaveguide with modulation gOf the capillary wall

(a)

(b)

Fig. 30 Shorter modulation periods of the capillary wall (a) extend the phase matching from 85 eV to 160 eV (b) in X-HHG by the Xe-filled capillary. (Pictures (a) and (b) from A. Paul et al., Nature (2Jan, 2003) and E Gibson et al Science (3 Oct 2003)


2003) and E. Gibson et al., Science (3 Oct. 2003),

Xe-plasma filled capillary waveguide extends phase matching from 95 eV to 150 eVphase matching from 95 eV to 150 eV

In case of a plasma-filled capillary waveguide, we have

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛−=

πλ

πλ

λπ 22

21

22112 eerN

auk (68)

⎥⎦⎢⎣ ⎠⎝⎠⎝

Fig. 31 Xe-plasma filled capillary waveguide extendscapillary waveguide extends phase matching from 95 eV to 150 eV. (Picture is from OPN, p.44, December 2006)


The use of Van Cittert-Zernike theorem for t d b X HHGx-rays generated by X-HHG

2R dOutput aperture of an x-ray source

Fig. 32 Schema for the use of theorem in case of an x-ray source base on X-HHG (2-Dimensional. circular, radius R) source composed from incoherent, uncorrelated Z

Θ

ybased on X-HHG

emitters

The X-HHG produces partially coherent or fully transversally coherent x-rays (waves) in the regionregion.

For a circular (radius R) surface of incoherent (uncorrelated) emitters of a HHG x-ray source, we have

ΘΘ

=Rk

RkJ )(2 112μ (69)

ΘRk

Thus the use of the Van Cittert-Zernike theorem for an x-ray source based on X-HHG yields



(70)(71)



(71)(72)

Spatial coherence of x-rays produced by X-HHG

z

Capillary hollow waveguide Laser wave

Wave

Two 50μmpinholesPhase shift

Wave by X-HHG

The 150 μm capillary filled with 30 TorrArgon is pumped by the ultra short (25 fs) laserArgon is pumped by the ultra-short (25 fs) laser Beam (λ=800 nm)

Fig. 33 Transverse (spatial) coherence of x-rays (λ=36 nm) produced by X-HHG(Fringes from Science 297, 376 (2002))


( g ( ))

Understanding vacuum tubes, synchrotrons, X FEL d X HHG i thX-FELs and X-HHG requires theory,

computations and experimentscomputations and experiments

Theory Computationsp

Experiment

Why can the 25-year theoretical and experimental experience of the University of Pecs in capillary plasmas and waveguides be useful for R&D of X-HHG?


Fig. 34 Understanding vacuum tubes, synchrotrons, X-FELs and X-HHG requires theory, computations and experiments

Problems as home assignments (A)1. Explain the continuum and characteristic radiation of an x-ray vacuum tube. 2. What is the shortest wavelength of the continuum x-ray radiation of an x-ray tube? 3. Explain the Van Cittert Zernike theorem for an x-ray vacuum tube.4 Give a physical explanation how relativistic electrons of a synchrotron produce x4. Give a physical explanation how relativistic electrons of a synchrotron produce x-

rays.5. Explain why relativistic electrons of a synchrotron produce x-rays in a narrow forward

cone.6 Explain the Van Cittert Zernike theorem for an x ray synchrotron6. Explain the Van Cittert Zernike theorem for an x-ray synchrotron.7. Give two important differences between 3rd generation storage rings and old rings.8. Describe important features of bending magnet X-FEL radiation.9. What are the important features of X-FEL wiggler radiation?10 Describe important features of X FEL undulator radiation10. Describe important features of X-FEL undulator radiation.11. What are the important advantages, even for users at modern storage rings?12.How is dipole radiation with relativistic transformations used to explain X-FEL

undulator radiation?13 Explain the Van Cittert Zernike theorem for X ray FEL13.Explain the Van Cittert Zernike theorem for X-ray FEL.14.How do the two factors of γ enter the undulator equation? 15.Explain the physical significance of each term in the X-FEL undulator equation.16.Why K is called the deflection parameter?17 What is the importance of the central radiation cone?17.What is the importance of the central radiation cone?18.Explain the dependence between the angular acceptance cone and the spectral

bandpass for X-FEL undulator radiation.19. How does the finite number N of magnet periods affect the acceptance cones and

the central radiation cone?


the central radiation cone?

Problems as home assignments (B)20. Explain the Van Cittert Zernike theorem for X-ray FEL.21. You are to design the X-ray source for a biological microscope (consult

[1]) ti i th t i d b t th b ti d f C d[1]) operating in the water window between the absorption edges of C and O. Assume you are working at a 1.5 GeV storage ring. What are your main considerations?

22. Select major parameters of the radiation source of the previous example.Y d i th t t d ti ti f i dYou are design the x-ray source to study magnetic properties of iron and

cobalt (consult [1]). What are your main technical considerations?23. What equipment do you need for the previous example?24. Draw a schema of your experiment. 25 E l i th V Citt t Z ik th f X HHG25. Explain the Van Cittert Zernike theorem for X-HHG.


References1. David Attwood, Soft-X-rays and Extreme Ultraviolet Radiation, Cambridge University

Press, 2000; David Attwood, Soft-X-rays and Extreme Ultraviolet Radiation (www.coe. berkeley.edu).

For additional information see: 1. R. C. Elton, X-ray lasers, Academic Press, 1990.2. David Attwood, Soft-X-rays and Extreme Ultraviolet Radiation, Cambridge University

Press, 2000; David Attwood, Soft-X-rays and Extreme Ultraviolet Radiation (www.coe. berkeley.edu).

3. J.J. Rocca, Review article. Table-top soft x-ray lasers, Rev. Sci. Instr. 70, 3799 (1999)4. H. Daido, Review of soft x-ray laser researches and developments, Rep. Prog. Phys.

65, 1513 (2002) 5. A.V. Vinogradov, J.J. Rocca, Repetitively pulsed X-ray laser operating on the 3p-3s

transition of the Ne-like argon in a capillary discharge, Kvant. Electron., 33, 7 (2003)6. S. Suckewer, P. Jaegle, X-Ray laser: past, present, and future, Laser Phys. Lett. 6,

411 (2009).


lecture 5 quantitative description of the xquantitative ... · an x-ray tube produces partially...

Documents